Answers to Questions by Dénes on Latin Power Sets

The i th power, Li, of a Latin square L is that matrix obtained by replacing each row permutation in L by its i th power. A Latin power set of cardinalitym ≥ 2 is a set of Latin squares { A, A2, A3, . . . ,Am }. We prove some basic properties of Latin power sets and use them to resolve questions asked by Dénes and his various collaborators. Dénes has used Latin power sets in an attempt to settle a conjecture by Hall and Paige on complete mappings in groups. Dénes suggested three generalisations of the Hall–Paige conjecture. We refute all three with counterexamples. Elsewhere, Dénes et al. unsuccessfully tried to construct three mutually orthogonal Latin squares of order 10 based on a Latin power set. We confirm as a result of an exhaustive computer search that there is no Latin power set of the kind sought. However we do find a set of four mutually orthogonal 9 × 10 Latin rectangles. We also show the non-existence of a 2-fold perfect (10, 9, 1)-Mendelsohn design which was conjectured to exist by Dénes. Finally, we prove a conjecture originally due to Dénes and Keedwell and show that two others of Dénes and Owens are false.